perspective projection

Perspective projection is the specific way the pinhole turns 3D into 2D: it is the rule that things farther away appear smaller, in exact proportion to their distance. This single effect — foreshortening — is what makes a flat image look three-dimensional, and it is why railroad tracks seem to meet at the horizon even though they are parallel. The mechanism is division by depth: an object twice as far away projects to half the size.

Precisely, a camera-frame point (X, Y, Z) maps to the image as x = f X / Z, y = f Y / Z, where f is the focal length and Z is depth along the optical axis. The defining feature is the Z in the denominator, which makes the mapping nonlinear in the world coordinates. A direct consequence is vanishing points: any set of parallel lines that are not parallel to the image plane projects to lines that meet at a single image point, the vanishing point, located in the direction the lines point. Lines that are parallel to the image plane stay parallel in the image.

Perspective contrasts with orthographic (parallel) projection, which simply drops the Z coordinate and preserves sizes regardless of distance — a good approximation for telephoto lenses or very distant, shallow scenes, and the standard model in some classic structure-from-motion work. Perspective is also not size-preserving or shape-preserving: a circle viewed obliquely projects to an ellipse, and a square tilted in depth becomes a general quadrilateral. Understanding which geometric quantities survive projection (incidence and straight lines do; lengths, angles and parallelism generally do not) is the entry point to projective geometry and to every calibration and reconstruction algorithm.